Application of Probabilistic Programming to the Multidimensional Data Compression Problem

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The contents of the researchers published during two years from 2002 till 2003 can be classified into two parts, which are satated as follows :(1).The solution to an open problem related to Hubert's 13th problem.The 13th problem formulated by Hubert, which asks if any continuous functions of several variables can be represented as superpositions constructed from several continuous functions of fewer variables, was solved by Kolmogorov and Arnold after about fifty years.Actually, theentire function theoretic problem asking if any entire functions of several variables can be represented as superpositions constructed from several entire functions of fewer variables has remained to be solved.The representative of this research project has succeeded in giving the solution to this open problem which is based on entropy theoretic methods.(2).A relation between the superposition representation proble'm and the theory of nomographs.Primarily, it has been shown that the concept of superposition irrepresentability can be classified. into two concepts which is called strong superposition irrepresentability and weak superposition irepresentability, respectively.Moreover, there does not exist any nomographs enabling to estimate the values of the functions of several variables if they are strongly irrepresentable, and, for a certain positive integer k, there does not exist any nomographs enabling to estimate the values of the functions of several variables by at least k-time operations if they are weakly irrepresentable.